Frequency, Central Tendency & Variability

Agenda

Frequency Distributions
Central Tendency
Variability

Readings

Being a Statistician Means Never Having to Say You’re Certain

\[ \definecolor{mathBlack}{RGB}{0,0,0} \definecolor{mathOrange}{RGB}{253, 126, 20} \definecolor{mathLightGreen}{RGB}{32, 201, 151} \definecolor{mathGreen}{RGB}{24, 188, 156} \definecolor{mathYellow}{RGB}{253, 156, 18} \definecolor{mathBlue}{RGB}{52, 152, 219} \definecolor{mathRed}{RGB}{231, 76, 60} \definecolor{mathPurple}{RGB}{111, 66, 193} \]

Frequency Distributions

FREQUENCY DISTRIBUTION
RELATIVE FREQUENCY DISTRIBUTION

PROPORTION
PERCENTAGE
CUMULATIVE
RATE

BAR GRAPH
HISTOGRAM
LINE GRAPH
STATISTICAL MAP

Objectives

Calculate proportions and percentages
Construct and analyze frequency, percentage, and cumulative distributions

DISTRIBUTION

Shows all the possible values (or intervals) of the data and how often they occur.

FREQUENCY DISTRIBUTION

A table reporting the number of observations falling into each category of the variable.

Tab A
Tab B
Tab C

Table 1. Attitudes about sex before marriage

premarsx	n
always wrong	357
almost always wrong	122
wrong only sometimes	258
not wrong at all	1,378
Total	2,115

Survey question: There’s been a lot of discussion about the way morals and attitudes about sex are changing in this country. If a man and woman have sex relations before marriage, do you think it is _________.

Table 1. Attitudes about sex before marriage

premarsx	n
always wrong	357
almost always wrong	122
wrong only sometimes	258
not wrong at all	1,378
Total	2,115

The number of respondents who answered this survey question.

Table 1. Attitudes about sex before marriage

premarsx	n
always wrong	357
almost always wrong	122
wrong only sometimes	258
not wrong at all	1,378
Total	2,115

The number of respondents who said pre-marital sex was “wrong only sometimes.”

Are women more likely than men to say premarital sex is “not wrong at all”?

Table 2. Attitudes about sex before marriage by gender

premarsx	male	female
always wrong	4,159	7,116
almost always wrong	1,499	2,388
wrong only sometimes	3,904	4,792
not wrong at all	10,672	11,086
Total	20,234	25,382

RELATIVE FREQUENCY DISTRIBUTION

A table showing the proportion or percentage for each value of a variable.

Proportions are between 0 and 1.0.

Proportion = count (f) / total number of cases (N).

Percentages are between 0 and 100.

Percentage = proportion × 100.

Tab A
Tab B

Table 2. Attitudes about sex before marriage by gender

	Men		Women
premarsx	n	%	n	%
always wrong	4,159	21	7,116	28
almost always wrong	1,499	7	2,388	9
wrong only sometimes	3,904	19	4,792	19
not wrong at all	10,672	53	11,086	44
Total	20,234	100	25,382	100

\(\frac{\color{mathGreen}{11{,}086}}{\color{mathOrange}{25{,}382}} = 0.4367 \times 100 = \color{mathRed}{43.7\%}\)

TIP: Total of a % column should always sum to 100!

Table 2. Attitudes about sex before marriage by gender

	Men		Women
premarsx	n	%	n	%
always wrong	4,159	21	7,116	28
almost always wrong	1,499	7	2,388	9
wrong only sometimes	3,904	19	4,792	19
not wrong at all	10,672	53	11,086	44
Total	20,234	100	25,382	100

A greater proportion of men (53%) than women (44%) say premarital sex is “not wrong at all.”

CUMULATIVE FREQUENCY DISTRIBUTION

The number or percentage of observations at or below a given category.

Table 3. Attitudes about sex before marriage, with cumulative percentages

premarsx	n	%	cumulative %
always wrong	357	17	17
almost always wrong	122	6	23
wrong only sometimes	258	12	35
not wrong at all	1,378	65	100
Total	2,115	100	175

\({\color{mathGreen} 17} + {\color{mathOrange} 6} = {\color{mathRed} 23\%}\)

RATES

\(\frac{Actual\;occurrences}{possible\;occurrences}\)

Examples:

Canada’s divorce rate decreased from 12.7 per 1,000 in 1991 to 5.6 per 1,000 in 2020.
The 2021 suicide rate of 14.8 per 100,000 population for middle aged Canadians (30-59 years old) was the highest of any age group.
Canada’s total fertility rate reached a new low in 2023 of 1.26 children per woman.

Nominal variables:
can have frequency distributions, cannot have cumulative frequency distributions

Ordinal:
can have frequency distributions and cumulative frequency distributions

Interval-ratio:
can have frequency distributions, cumulative frequency distributions, and rates

A bar graph is used:
for nominal or ordinal variables,

to show frequencies or percentages,

using separated rectangles, with height proportional
to the frequency or percentage.

A histogram is used:
for interval-ratio variables,

to show frequencies or percentages,

using separated rectangles, with height proportional
to the frequency or percentage.

A line graph is used:
for interval-ratio variables,

to show frequencies or percentages,

joining by category the frequency or average with a line.

A statistical map is used:
for interval-ratio variables,

to show geographical variations, often in ratios,

using variation in color or hue.